Low frequency asymptotics for the reduced wave equation in two-dimensional exterior spaces

We consider the Dirichlet problem for the reduced wave equation ΔU_x + x²U_x = 0 in a two-dimensional exterior domain with boundary C, where C consists of a finite number of smooth closed curves C₁,…,C_m. The question of interest is the behavior of U_x as ? → 0. We show that U converges to the solution of the corresponding exterior Dirichlet problem of potential theory if the boundary data converge to a limit uniformly on C. This generalizes a well-known result of R. C. MacCamy for the case m = 1.